## Volume 458,   Issue 48,   1999

• Sklyar G. M., Skoryk V. A. On a set of positional controls solving the sysnthesis problem in the Hilbert spaces , P. 3-14. [ Abstract ]
• Poslavsky S. A., Miller H. A. On a calculation method for convergent shock waves , P. 15-21. [ Abstract ]
• Lysenko Ju. V., Ovcharenko I. E. On Algorithmic questions of the inverse and direct problems for Stiltjes's string and orthogonal polynomials , P. 22-42. [ Abstract ]
• Sklyar E. V. The determination of the explicit form of the control and trajectories solving the controllability for certain non-linear systems , P. 43-52. [ Abstract ]
• Kantor B. J., Naumenko V. V., Strelnikova E. A. Indirect formulation of Boundary Element Method in 2D elasto-plastic problems , P. 53-61. [ Abstract ]
• Zagorodniuk S. M. On polynomial sequences in Hilbert space , P. 62-78. [ Abstract ]
• Suzikov G. V., Vorobyov I. V., Rozumenko O. V. About one method of the solving the N-th order boundary problem , P. 79-90. [ Abstract ]
• Marchenko I. I. On Wiman-Valiron theorem , P. 91-93. [ Abstract ]
• Levitskaya T. I. The Usage of Superelement Method for Stress-Strain Calculation of the Shell of RevolutionMarchenko I. I. , P. 94-100. [ Abstract ]
• Piven A. L. Solvability of Cauchy problem and estimates of initial manifold for a nonimplicit operator differential equation , P. 101-108. [ Abstract ]
• Chernovol N. Many--dimensional triangular models of linear operators systems with nilpotent commutators , P. 109-118. [ Abstract ]
• Zolotaryov V. A., Vorobyov I. V. Normal metrical knot , P. 119-129. [ Abstract ]
• Lutsenko A. V., Skoryk V. A. Global null controllability of linear discrete systems , P. 130-142. [ Abstract ]
• Dyukarev Yu. Œ. The multiplicative structure of resolvent matrix of interpolation problem in Stieltjes class , P. 143-153. [ Abstract ]
• Tseytlin L. M. Notes on Vector Integration , P. 154-158. [ Abstract ]
• Khudoshin I. G. Initial value problem for some quasilinear differential-algebraic equations. , P. 159-164. [ Abstract ]
• Popova H.V. Existence and uniqueness of basic solutions for electrodynamical system. , P. 165-176. [ Abstract ]
• Malyutina T. I. Asymptotic expansions of integrals with nonregular growth , P. 177-184. [ Abstract ]
• Legeyda D. V., Tarapov I. Ye. The flat MHD-bearing with an injection or suction on walls and distributed source of mass , P. 185-193. [ Abstract ]
• Ahiezer E. B. On inhomogeneous vector fields in Hilbert spaces , P. 194-204. [ Abstract ]
• Cheremskaja N. V. Nonstationary sequences generated by differential-difference equations in Hilbert space , P. 205-212. [ Abstract ]
• Chudinovich I. Boundary equations in dynamic problems for thin plates , P. 213-217. [ Abstract ]

The solution of the positional control synthesis problem for a controllable process described by a linear differential equation with bounded operators in Hilbert space is considered. Namely, a set of restricted positional controls such that all solutions of the corresponding equation with an arbitrary initial condition tend to zero for certain finite time is proposed. The constructive solution of this problem based on the controllability functional method is obtained. It is shown that any non-increasing non-negative function on the semiaxis $[0,\infty)$ having sufficiently large number of the points of decreasing generates a control solving the considered problem. [ Top of the page]

A new application of the well-known approximate method of Chester-Chisnell-Whitham for calculation of shock waves evolution is considered. It is shown that within this method it is possible to reduse the non-stationary problem of wave front spreading to the problem of steady supersonic gas flow with a specially chosen polytropic exponent. Applying hodograph variables made it possible to obtain the solution in an analytical form. Numerical calculations were carried out for a convergent shock wave having an initial form of an elliptic cylinder. [ Top of the page]

The article offer step by step algorithms, based on Chebyshev's recursion for the recovery of the Stiltjes's string from the moments of its spectral functions and step by step algorithms to test whether the moment sequence is a positive definite in Stiltjes's sense. [ Top of the page]

In the work the controllability problem for a special class of the triangle systems is considered. The main result is the determination of the explicit form of the control and trajectories realizing movement from a given point to another one. The both cases of the control without restrictions as well as the restricted control are investigated. [ Top of the page]

The method of solution the boundary value problems for non-elastic bodies is supposed. The indirect formulation of boundary element method is in use. To determine the elasto-plastic stress and strain fields the iterations method is applied. At each iteration the linear boundary value problem is solved, that is described by the system of singular integral equations. The numerical investigation of elasto-plastic stresses concentration in the bodies with holes was done. The unique solvability is proved of systems of boundary equations that emerge in oscillation problems for thin elastic plates with boundary data of a mixed type. [ Top of the page]

We study the properties of polynomial sequences in separable Hilbert space. The criterion for arbitrary sequence to be polynomial is given. We consider sequences corresponding to classical orthogonal polynomials. Ergodic formulas, decomposition for representation measure in series are investigated. [ Top of the page]

The paper studies the method of the solving the boundary problems for the N-th order ordinary differential equation, which is that the solution of the "difficult" boundary problem is represented in the form of the limit of the solutions of the "simple" problems -- the Cauchy problems for the same equation. The main result, in fact, is that the addition of one cycle in the program of the solving the Cauchy problem for the differential equation with the simple recalculation after every passing, transforms it in the program of the solving the boundary problem. The method is very simple and effective, has an extensive domain of employment. [ Top of the page]

Influence of a number of separated points of maximum of modulus of an entire function on the circle $\{z: |z|=r\}$ on Wiman-Valiron inequality is investigated. [ Top of the page]

The usage of superelement method for composite shells of rotation enabled to work out an effective algorithm of calculation of their stress-strain state. For the element of ring-type the local stiffness matrix has been made, thus giving the possibility to observe shell-type ribs as well as ring-type ones, which cross-section sizes are smaller than their radius. A number of calculations have been made to compare the impact of the rib form upon the tense state of shells. [ Top of the page]

In this paper we consider the Cauchy problem $\sum\limits_{j=0}^{n}A_ju^{(j)}(t)=0,\ t0;\ u^{(j)}(0)=u_j,\ j=0,...,n-1$ with linear closed operators $A_j$ from a Banach space $X$ into a Banach space $Y$. The operator $A_n$ has, in general, a non-trivial kernel. By means of a solvability theorem for this Cauchy problem the initial manifold is estimated and the conditions of the its non-triviality are established. The example of the mixed problem satisfying to the proved theorems for some partial differential equation is considered. [ Top of the page]

A triangular model a completely not self-adjoint system of linear bounded operators is constructed. It concerns of acting in Hilbert space H operators $A_1$ and $A_2$ such that their commutators $D=[A_1,A_2]$ and $C=[A_1^*,A_2]$ are nilpotent ones. A special class of operator systems $A_1$, $A_2$ is extracted. It result in the model realisation, where $A_1$ and $A_2$ are operators of integration with respect to independent variables $x_1$ and $x_2$ (respectively) in space $L^2(\Omega)$, here $\Omega$ is a domain in $R^2$, that is defined by the properties of nilpotent commutators $D$ and $C$. [ Top of the page]

Operator knots allow to study the spectral properties of the different families of operators. Such knots were created for the nearly self-adjoint and nearly unitary operators. In the given paper we present the model of a such knot for the nearly normal operators and show, that the characteristic function of the knot is an unitary invariant of these knots, and also describe a class of functions, which are the characteristic functions. [ Top of the page]

The paper considers the problem of steering one state of a linear time-varying system, discrete-time system to the origin, when control is subject to magnitude constraints. Necessary and sufficient conditions are given for global null controllability by means of as uncontrained controls as contrained controls. Sufficient conditions are given for global null controllability of perturbed sustem. [ Top of the page]

In this paper a special representation of resolvent matrix of interpolation problem in Stieltjes class is introduced. The multiplicative structure of resolvent matrix is study and some examples are given. [ Top of the page]

The article announces some results of the author's Ph.D. thesis and introduces a new definition of an integral for functions with values in Banach spaces. This definition, called Riemann--Lebesgue integral, is shown to be quite general (both Bochner and Riemann integrals are its special cases) and useful in the stufy of Banach spaces. [ Top of the page]

In this paper some conditions of solvability of equation $A\frac{dx}{dt} + Bx(t) = f(t,x), a \leq t \leq b$ in Hilbert spaces are obtained in case of a pole of the resolvent $(\lambda A+B)^{-1}$ in the point $\lambda=\infty$ and finite-dimensional root subspace $X_{2}.$ Representation of spectral projectors via biorthogonal chains of eigen and associated vectors of initial and conjugate pencils is obtained. [ Top of the page]

Non-monochromatic fields in cylindrical waveduct with nonhomoheneous nonstationary medium are expanded over countable system of specialized modal fields. The conditions for existence and uniqueness of basic solutions for a mixed problem of Maxwell's equations are presented. [ Top of the page]

The integrals $$\int_a^bf(t)\exp(i|\ln rt|^\sigma)\,dt.$$ are investigated. Asymptotic properties of ones are different in cases $\sigma\in(0,1)$, $\sigma=1$, $\sigma\in(1,\infty)$. These integrals have nonregular growth. There is an analogy with the classic Fourier integrals if $\sigma1$. [ Top of the page]

The influence of an injection (suction), and also extended sources of a mass on performances of the flat MHD-bearing is researched. In an approximation of the hydrodynamic theory of lubrication the distribution of a velocity on cut of a lubricating stratum is retrieved and are calculated force of friction on walls and body force. The graphs of association of a factor effectivensses of the bearing from Alphen number, velocity of a having blown, declination of a jet of an injection are obtained. The influence of radiants and drains of a mass on flow of a liquid and effectiveness of the bearing is installed. [ Top of the page]

In the paper inhomogeneous vector fields in Hilbert spaces are studied. The concept of evolutionary representable vector fields is introduced and the criterion of evolutionary representability in terms of a correlation matrix is given. The infinitesimal correlation matrix which characterizes a deviation of the field from the homogeneous one is defined. The theorem of the finite rank for evolutionary representable vector fields is proved. [ Top of the page]

In the paper nonstationary sequences in Hilbert spaces are studied with the help functional equations and triangular models of nonselfadjoint operators. The evolution of space-time spectrum are studied. [ Top of the page]

The unique solvability is proved of systems of boundary equations that emerge in oscillation problems for thin elastic plates with boundary data of a mixed type. [ Top of the page]

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Visnyk Kharkivs'koho natsional'noho universytetu imeni V. N. Karazina, Seriya «Matematyka, prykladna matematyka i mekhanika»

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